Modeling Multiple Peer Influences on Repeat Purchase by Online Game
Players
Ruibin Geng
Zhejiang University
Bin Zhang
University of Arizona
Xi Chen
Zhejiang University
The bulk of the work was done by a student
Abstract
Consumers’ online purchase decisions can be affected by both direct and indirect peer influences.
Literature has provided empirical evidence about both types of influence when the outcome is
binary. However, there is no existing work about how these two types of peer influence impact the
repeat purchase of consumers. In this study, we want to fill such gap by investigating the
interdependent repeat purchase of online game players embedded in a social network. We build a
new hierarchical Bayesian model that can support response variable following Poisson
distribution and simultaneously include both direct and indirect influences. We compared the
Bayesian analysis of the spatial panel models with fixed and random effects. Empirical results
show that both types of peer influence have significant impact on players’ repeat purchase but the
two influences have different directions. Also, we find that the size of the peer influences is
associated with both network structure features and the strength of social pressure. The
implication of our newly developed model includes first, expanding IS literature by studying the
diffusion of repeat purchase for virtual goods, and second, helping online game providers develop
effective targeting strategy to convert more purchase through peer influence.
Keywords: Online games, Peer influence, Cohesion, Structural equivalence, Poisson regression,
Network autocorrelation, Hierarchical Bayesian model
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1. Introduction
Literature shows that individuals’ purchase decision can be affected by peer influence (Leenders
1997). For example, decisions about purchasing applications on Facebook website (Aral and
Walker 2011), caller ringback tone on mobile phone platform (Ma et al. 2014), and iPhone 3G
(Godinho de Matos et al. 2014) etc. can all be influenced by neighbors in social networks. However,
there are two types of peer influence that provide very different explanations. The first type is
direct influence, or cohesion. It explains that peer influence only comes from the direct neighbor
because the impact has to be made through direct communication (Coleman et al. 1966). The
second type of influence is called as indirect influence, or structural equivalence. Such influence
occurs when individuals imitate indirect connected actors with whom they share common
neighbors (Burt 1987).
These two explanations about peer influence create two camps among scholars, and both
camps have found empirical evidence to support their statements. Coleman et al. (1966), Rogers
and Kincaid (1981), and Harkola and Greve (1995) demonstrate that individual’s adoption decision
is affected by direct influence, while Burt (1987), Strang and Tuma (1993), and Van den Bulte and
Lilien (2001) find that adoption diffusion among individuals within the social network is driven
by indirect influence. Bowler et al. (2011) reviewed the papers that examine whether social
contagion effect is a result of relational adjacency (direct influence) or a similarity of network
position (indirect influence). They conclude that direct and indirect influences have independent
and simultaneous impact on adoption behavior. The debate between direct (cohesion) and indirect
influences (structural equivalence) as the explanations of behavioral conformity remains
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ambiguous. The problem that this manuscript wants to settle is verifying these two different
explanations, direct (cohesion) and indirect influence (structural equivalence), in the context of
repeat purchase by online game players embedded in a big social network. Specifically, we study
which type of peer influences, or both types, has impact on the counts of purchasing virtue goods
by game players connected by the in-game social network. Figure 1 illustrates our basic
assumption that the high frequency purchase behavior of focal player will induce the increase of
repeat purchase of her social neighbors and of individuals who share common friends with her via
direct and indirect influences.
[Insert Figure 1 around here]
To our knowledge, our work is the first to study peer influence on individuals’ repeat purchase.
Traditional models for consumer repeat purchase, such as Pareto/NBD model (Schmittlein et al.
1987) and its hierarchical Bayes extension (Abe 2009), both ignore the interdependence of
decisions by individuals within social networks. These two models address the heterogeneous
preferences across individuals either by exogenous covariates or by independent and identical
draws from a mixing distribution (Rossi and Allenby 2003). For literature studying the impact of
peer influence on purchase decision (Yang and Allenby 2003; Zhang et al. 2013; Fang et al. 2013),
only the circumstances of binary choice (purchase product or not) and continuous outcome
(amount of payment) are investigated. Since the price distribution among virtual goods is highly
skewed, and heavily determined by supply and demand sides in the game. It makes price for virtual
goods change frequently and drastically, thus the amount of payment by players does not
necessarily represent their consistent willingness to pay in a long time period. In terms of binary
3
decision models, the first adoption behavior does not guarantee that the individual will continue
purchasing the product in the future. Using a count type data, we are able to monitor the fluctuation
of players’ repeat purchase behavior. Game operator would be more interested in targeting players
having high sustainability in the game, compared to those who purchase one expensive virtual
goods but play much less and buy much less afterwards. Therefore, a model that can support the
number of repetitive purchase made by players may be more useful than models supporting of
binary or numeric outcomes.
In this study, we want to fill such gap by investigating the interdependent repeat purchase of
online game players embedded in an in-game social network. We build a new hierarchical
Bayesian model that can support response variable following Poisson distribution and
simultaneously include both types of peer influences, direct and indirect. Specifically, the direct
and indirect peer influences are modeled as two network autocorrelation terms, which capture the
correlated unobservable heterogeneity across individuals. We applied panel data to our Bayesian
model, which extends the modeling possibilities and increase the efficiency of estimation as
compared to the cross-sectional setting (Elhorst 2014). Our empirical results show that both types
of peer influences have significant impact on players’ repeat purchase decision but the directions
of the two influences are different. Furthermore, we find that the size of the peer influences is
associated with both network structure features, such as degree and betweenness centrality, and
the strength of social pressure. With regard to practice implications, our model helps online game
providers develop effective targeting strategy to convert more purchase through peer influence.
4
The reminder of this paper is organized as follows. Section 2 describes the data and research
context. Section 3 provides the model specification and estimation procedures. Section 4 illustrates
the empirical results. Section 5 and 6 offer a discussion and conclusion.
2. Data and Context
Our data is obtained from one of the largest online game operators in Asia about a massively
multiplayer online role-playing game (MMORPG). Such game adopts a freemium model (Riggins
2003) – it is free-to-play but the advanced virtual goods are charged with prices. These virtual
goods are purchased to enhance the ability of the character used by the player or to customize the
character for decoration purpose (Guo and Barnes 2011). In order to proceed in the game, players
need to repetitively buy virtual goods and to team up with other players to accomplish tasks.
Different from that in the real world, players cannot buy virtual items directly with real money in
online games. They need to purchase in-game currency first using real money, then use the virtual
currency to obtain virtual items. Determined by the nature of the freemium model, we only need
to investigate the purchase of in-game currency, while the purchase of virtual items using in-game
currency does not fall within the scope of our study. We collected the repeat purchase of 12,220
players over a 12-week period in the game. Although these paid players only account for a small
proportion of the whole player base, they generate the majority of revenue of the online game.
Thus understanding the factors that affect players’ continuous purchases and sustain these
purchases is of great importance to online game operators.
In order to finish the tasks in the game, characters need to build up teams. Thus the game
provides in-game social networking service to facilitate the team building. For example, players
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can create mutual friend relation with each other. Characters can also join ‘guilds’, an organized
group of characters who regularly play together, usually with a common interest such as finishing
in-game tasks and sharing trophies. Both the friend relationship and the guild membership may
affect players’ repeat-purchase behavior. On the one hand, characters equipped with expensive and
powerful items may motivate their friends to spend more money and finish more challenging tasks.
On the other hand, guild members (characters) are required to donate to the guild funds (in-game
currency) in order to remain in the guild, which may cause a herding effect on repeat purchase
among the members.
The creation of the friend relation is based on mutual agreements between two characters,
making the relationship reciprocal. Consequently we defined the in-game friend network as
undirected. Our data is aggregately collected from four distributed servers with independent copy
of the game. There is no significant difference in the players’ personal attributes, such as
geographical location, among these four servers. In large networks, individuals intend to construct
self-contained groups (subnetworks) with dense internal connections and sparse external
connections. The definition of the subnetwork described above is very similar to that of a
community, as in community detection. We hence apply the Louvain method (Blondel et al., 2008)
to extract the subnetworks whose nodes are densely connected, so that we are able to identify the
unique pattern of peer influence contained in each subnetwork. As shown in Figure 2, there exist
four subnetworks in the global network. The network structural properties of these four
subnetworks are demonstrated in Table 1.
[Insert Figure 2 around here]
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[Insert Table 1 around here]
In the existing literature, there are three categories of factors, personal attitudes, products
attributes, and the dynamics of the environment, are associated with virtual product consumption
(Chung 2005; Lehdonvirta 2009; Joo et al. 2011). According to the literature about product
consumption decision, repeat purchase is determined by the product attributes when there are
multiple products existing (Kim et al. 2011). However, our research focuses on the repeat purchase
of one single product, the virtual currency in the game, rather than the heterogeneous virtual items
that can be purchased. Therefore, we only include personal attributes and social factors, such as
peer influence from the friend network and herding effect from the guild, in our model. Based on
our context and prior studies, we collect individual and guild attributes that may impact the repeat
purchase in online games. Individual attributes include level, online duration, login count, and
count of quests completed etc.. Guild attributes that may exert group pressure on herding behavior
of repeat purchase include number of paid members in the guild, guild level, guild money and a
leader flag etc.. In terms of peer influence, it can be interpreted as the correlation among
individuals coming from observed behavior or latent preference (Leenders 2002; Zhang et al.
2014). In online games context, the in-game purchase behavior of player is unobservable to others
including her friends. Thus, the peer influence in our model can only be reflected by the correlated
structure of latent preference across individuals.
3. Model
So far existing models studying peer influence only support continuous or binary response
variables (Coleman et al. 1966; Leenders 2002; Zhang et al. 2014; Fang et al. 2013). Much less
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models are available to accommodate count variable, when investigating multiple types of peer
influences. To our knowledge, we are the first to design a Poisson regression model with two
network autocorrelation terms that capture both direct and indirect peer influences among
individuals. We aggregate the data using a two-week time window to construct a balanced panel
data, which allows unobserved time-invariant individual effects to be defined as either fixed or
random effects. Such time-invariant effects cannot be captured using pure cross-sectional data
(Hsiao 2005). The use of panel data increases the estimation efficiency and enhances the
identification in causality. The matrix form of the model is specified as below:
y t ~ Poisson(λ t )
log(λ t )  Xt β  θt +η  ε t , ε t ~ N (0, I n )
θt  1W1θt   2 W2θt  ut , ut ~ N (0,  2I n )
β ~ N (β 0 , D),  2 ~ IG ( s0 , q0 )
yt is the vector of repeat-purchase counts of individuals through a certain time period. It
follows a Poisson distribution specified by the purchase rate parameter 𝛌𝒕 that varies across
individuals. Purchase rate parameter 𝛌𝒕 can be represented as a function of covariates 𝐗 𝒕 , timeinvariant individual effects 𝛈, and network autocorrelation term 𝛉𝒕 . The covariate matrix 𝐗 𝒕 are
defined by players’ individual attributes and guild-level attributes. We also add degree centrality
into 𝐗 𝒕 as control variables in the regression equation. 𝛃 is the correspondent coefficient for 𝐗 𝒕 .
A key issue in panel data modeling is to distinguish fixed or random effects by identifying whether
correlation between the individual effects 𝛈 and regressors 𝐗 𝒕 is allowed. In our context, there
exist several unobserved personal attributes that are time-invariant and may affect the repeat
purchase rate 𝛌𝒕 . Some unobserved personal attributes, for instance, the degree of addiction to the
8
game, may be correlated with the measured regressors, however, some other attributes, such as,
sex or education background, can be uncorrelated with the regressors. Thus we provide two
different assumptions on the individual effects term 𝛈 to accommodate flexible situations. In
classical econometrics, fixed effects are treated as parameters and estimated by a least-square
dummy variable estimation, while random effects are assimilated to the error term and estimated
by a generalized least squats. However, in the Bayesian framework, both fixed and random effects
are defined as random parameters with different prior assumptions. In fixed effects model, 𝛈 is
assumed to be correlated and follow a certain multivariate normal distribution. In random effects
model, 𝛈 is treated as independently and identically distributed random variable. The fixed
effects estimation updates the distribution of the parameters, while the random effects estimation
updates the distribution of the hyperparameters (Rendon 2013). Our assumptions of fixed and
random individual effects 𝛈 are illustrated below:
fixed effects: η~N (μ0 , Γ0 )
random effects: η~N (0 , 2 ), 0 ~ N ( 0 ,  20 ), 2 ~ IG(A, B)
We captured two peer influences, direct and indirect influences, in the autoregressive
parameter 𝛉𝒕 correspondingly represented by two network autocorrelation terms 𝜌1 𝐖1 𝜽1 +
𝜌2 𝐖2 𝜽2 . The scalar 𝜌1 and 𝜌2 are coefficients for 𝐖𝜽. The two weighting matrices respectively
as adjacency matrix and inverse euclidean distance matrix as below:
W1  {aij }, where aij {0,1}
W2  {
1
}, where dij   zn \{i , j} (aiz  a jz ) 2
dij  1
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The main diagonal entries of the two matrices are equal to zero, so no self loop is permitted.
The entry 𝑎𝑖𝑗 in matrix 𝐖1 equals one if i and j are mutual friends and zero otherwise. The entry
in matrix 𝐖𝟐 specifys the power of structural equivalence between every two players, which is
calculated by the inverse of Euclidean distance 𝑑𝑖𝑗 plus one (Zhang et al. 2013). The higher value
of the distance, the less structural equivalence between the two nodes.
We use Markov chain Monte Carlo (MCMC) method to solve the Poisson regression model
with network autoregression terms. The basic idea is to proceed a Markov chain by generating
samples from the set of posterior distributions successively. The description of MCMC estimation
implementation is shown in Appendix A and B.
4. Results
We estimated our hierarchical Bayesian model using MCMC methods. The first 2,000 draws
were discarded as “burn-in” period, and then we ran an additional 8,000 draws. The estimation
sequence was thinned by keeping every 20th simulation to avoid autocorrelation problems among
MCMC iterations. Estimation results based on the fixed and random effects models are shown in
Table 2.
[Insert Table 2 around here]
For model selection, we use the deviance information criterion (DIC) (Spiegelhalter et al.
2002) to decide whether the time-invariant individual effects are either fixed or random. DIC is
the most common method of assessing the goodness of fit in Bayesian inference. The model with
the smallest DIC is estimated to be the model that best predict a replicate dataset which has the
10
same structure as the currently observed. According to Table 2, the mNAPO model with fixed
effects outperform the random effects one in goodness of fit to the data.
In-game characteristics of players include both positive and negative driving factors.
Achievements in the game, accumulated playing duration, and the login frequency all provide
positive stimulus to the repeat purchase of players. But the level and quest accomplished frequency
that represent the in-game skill of players are examined as negative signal of repeat purchase
behavior. The results are reasonable because all the three positive stimuli indicate the degree that
the player is addicted to the game whilst the other two negative signals indicate that only less
skilled game players have more incentive to constantly invest money. In addition, the influence of
group level attributes on individuals’ repeat purchase behavior is only significant in subnetwork 3
and 4. These group-level attributes illustrate the compliance pressure from group regulations or
collective strategies. The results indicate that only the players in subnetwork 3 and 4 receive
significant guild pressure to make payments.
Peer influences which are indicated by parameters 𝜌1 and 𝜌2 are the key elements of
interest in our model. As we have defined, 𝜌1 represents the direct peer influence, while 𝜌2
addresses the indirect peer influence. The overall effects (in Table 2) of direct influence in
subnetworks 1, 2, 4 are significantly positive, which is in accord with the expectation that the
behavior of neighboring players are positively correlated. The effect of direct influence is not
significant in subnetwork 3 which is the most densely connected subnetwork among the four. On
the other hand, indirect influence has negative effect on the repeat purchase and it is only
significant in subnetworks1 and 4, in the opposite from our expectation.
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Peer influences play quite different roles in the four subnetworks as illustrated above. It has
been proved that network structure properties, such as network density and centrality, etc.,
associate with the size of peer influence in social networks (Haynie, 2001). So we conduct
ANOVA analysis on network structure features at individual level, including degree and
betweenness centrality, across the four subnetworks. The ANOVA analysis demonstrate that the
average degree of players in subnetwork 1 is significantly smaller than that of the other 3
subnetworks and the average betweenness centrality of players in subnetwork 1 is significantly
larger than the others. Thus we propose that direct peer influence is more likely to take strong
effects in lower average degree centralized subnetwork provided that the network is connected
with no isolated nodes, while indirect peer influence is positive associated with the average
betweenness centrality of the subnetwork. However, when only control for average degree and
betweenness centrality, the fact that direct influence in subnetwork 4 is weaker than that in
subnetwork 2 cannot be explained. We can also realize that the group-level influence on purchase
behavior is only significant in subnetwork 3 and 4. It suggests that the direct influence in
subnetwork 4 is partial substituted by group-level influence. The external pressure from the guild
may reduce the behavioral dependency between in-game friends.
5. Discussion and Contribution
5.1 Discussion of Findings
Our study has three notable findings. First, we empirically show that repeat purchase made
by players is determined by the interdependent latent preference of individuals in the network,
even though the purchase behaviors of the neighbors are unobserved in the network. Meanwhile
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the preference could be affected by multiple types of peer influence simultaneously. Second, we
find that both direct and indirect peer influences may exert significant impact on players’ purchase
behavior but the directions of effects are different. Third, the size of the two peer influences is
associated with network structural properties as well as the strength of social pressure from herding
effect. Overall, peer influences exhibit a very deterministic role in driving repeat purchase.
Since the individual purchase behavior, such as buy or not and what being bought, is
unobserved to other players in the network, in our context peer influence cannot be interpreted as
behavior imitation, which is usually suggested by traditional literature. The estimation results of
our model provide the evidence that there exist multiple types of peer influence among game
players, and such peer influences affect the repeat purchase behavior of focal individual through
latent preference dependency from other players.
Another crucial finding is that although direct and indirect peer influences may exert
significant influences on repeat purchase, the directions of the effects are different. The positive
correlation of latent preference among directly connected players is in accord with the expectation.
However, indirect peer influence (structural equivalence) is not always significant and has the
opposite sign from expected direction. Previous literature on the effect of indirect influence
(structural equivalence) provides mixed evidence as well. The empirical works of Mizruchi (1993)
and Zhang et al. (2013) find negative indirect peer influence on political behavior among
competing firms and on subscription decision on ‘Caller Ring Back Tones’ services. Zhang et al.
(2014) point out that negative indirect peer influence may be observed when individuals intend to
differentiate themselves from individuals who share common friends with them. In the game
13
context, purchasing behavior of player usually occur as a compensation action for lacking of skills
and experiences in games. Players’ investment decisions will be protected from imitation by
indirect connected individuals sharing common friends. So insignificant indirect peer influence
and even negative correlation among indirect connected players sharing common friends are
observed in our studies.
The third finding is that the size of both types of peer influences in each subnetwork is
associated with the average degree and betweenness centralities of the correspondent subnetwork.
For degree centrality, its mean value in the subnetwork is negatively correlated with the size of
direct peer influence. At the first glance, this finding may be counterintuitive that peer influence
will be enforced in social network with high density or average degree level. As it is defined,
degree is the number of one-hop friends the individual has. Within a connected network, the lower
average degree is, the more efficient information diffuses. Thus, the direct peer influence through
each social tie is more likely to take strong effects. Similar results have been found by Ugander et
al. (2012) that the size of neighborhood generally has negative effect on social contagion. For
betweenness centrality, its mean value in the subnetwork is positively associated with the size of
indirect influence. Betweenness centrality is used to measure the influence power from social
position. Betweenness centrality quantifies the number of times a node acts as a bridge along the
shortest path between two other nodes, which addresses the important role of common friends in
the concept of structural equivalence (indirect influence) (Rupnik 2006). Thus, a high value in
average betweenness of all nodes indicates strong impact from indirect peer influence. However,
subnetwork 4 shows inconsistent results that the size of direct peer influence in subnetwork 4 is
14
smaller than that in subnetwork 2, although the average degree centrality of subnetwork 2 is higher
than that in subnetwork 4. We find that in a subnetwork, if the group-level influence on purchase
is strong, the direct peer influence within the subnetwork will be weakened. This result can be
explained by the substitution effect between blended relationships in previous studies about the
impact of diversified relationships in organization on behavioral processes (Ragins 1997). Peer
influence and herding effect from the guild (crowd) are two substitutive powers to drive the repeat
purchase (Lee et al. 2015).
5.2 Implication to Methodology and Literature
We make contribution in methodology by creating a new Poisson regression model with two
network autocorrelation terms that represent the impact of both direct and indirect peer influences
on repeat purchase behavior of online game players. The augmented-error model contributes to
nonzero covariance of purchase rate 𝛌 , which sheds light on two possible sources of the
interdependent preferences from relational and positional social influence perspectives. We use
the longitudinal model for panel data to identify the causality that the change in the repeat purchase
behavior is caused by the change in players’ personal attributes and peer influences, and hence
increases estimation efficiency as compared to cross-sectional data.
Our study expands IS literature by studying the diffusion of repeat purchase for virtual goods
in online games. The frequency analysis of behavior is able to detect the non-progressive
phenomena, in which an individual may start using or purchasing a product and then possibly stop
at some point. The initial adoption behavior does not guarantee that the individual will continue
using or purchasing the product in the future. Using a count type data, we are able to monitor the
15
fluctuation of players’ repeat purchase behavior. In addition to games, our model can be applied
to diversified count data scenarios. For instance, our model can be applied to study the frequency
of login in social media, the frequency of content creation and consumption in user-generatedcontent-based websites, etc., which will provide broader insights for diffusion theory.
5.3 Implication to practitioners
Our work can also provide implications to practitioners, especially to online game operators.
First, our model has an effective capability to predict players’ future transactions based on the
estimated value of purchase rate in Poisson distribution. In addition, the Bayesian estimate of
purchase rate for each player helps online game providers and operators develop effective targeting
strategy to convert customers purchase more through peer influence. Second, the regression part
provides a comprehensive set of key factors underlying players’ repeat-purchase behavior. Our
findings suggest that managers should take advantage of positive peer influence through direct
social connections and regulate the information asymmetry between indirect connected players
sharing common friends. Advertising and promotion efforts should be invested on influential
players and subnetworks with effective network structures.
6. Conclusion
We design a new hierarchical Bayesian model to identify the key factors that motivate online
game players’ repeat-purchase behavior. Empirical results indicate that both direct and indirect
peer influences can have impact on repeat purchase simultaneously. Properly identifying such peer
influence mechanism provides substantial may benefit to effective future transactions prediction.
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Yang, S., & Allenby, G. M. (2003). Modeling interdependent consumer preferences. Journal of
Marketing Research, 40(3), 282-294.
Zhang, B., Susarla, A., & Krishnan, R. (2014). A Bayesian Model to Predict Content Creation
with Two-Sided Peer Influence in Content Platforms. Available at SSRN.
Zhang, B., Thomas, A. C., Doreian, P., Krackhardt, D., & Krishnan, R. (2013). Contrasting
multiple social network autocorrelations for binary outcomes, with applications to
technology adoption. ACM Transactions on Management Information Systems (TMIS),
3(4), 18.
Figures and Tables:
Figure 1. social contagion process on repeat purchase of online game players
20
Figure 2. Structure of friend network in the game, consisting of four subnetworks
Table 1. Network structural properties of the four subpopulations
Network
Network
Network
Average
Average
Size
Density
Diameter
Degree
Betweenness
Subnetwork 1
3045
0.003
10
9.616
4190
Subnetwork 2
2100
0.006
8
11.64
2615
Subnetwork 3
2082
0.006
8
12.55
2528
Subnetwork 4
2531
0.004
8
11.11
3271
21
Table 2. Estimated coefficients from the Bayesian model with fixed and random effects
fixed effects
random effects
subnetwo
subnetwo
subnetwo
subnetwo
subnetwo
subnetwo
subnetwo
subnetwo
rk 1
rk 2
rk 3
rk 4
rk 1
rk 2
rk 3
rk 4
Achievement
0.160**
0.204**
0.171**
0.168**
0.231**
0.245**
0.239**
0.229**
(𝛽1 )
(0.022)
(0.025)
(0.026)
(0.023)
(0.017)
(0.020)
(0.019)
(0.018)
-0.008
-0.012
-0.006
0.021**
-0.009**
-0.005**
-0.002
-0.009**
(0.008)
(0.011)
(0.012)
(0.008)
(0.001)
(0.002)
(0.002)
(0.002)
0.434**
0.405**
0.365**
0.293**
0.223**
0.175**
0.155**
0.074**
(0.030)
(0.034)
(0.036)
(0.031)
(0.017)
(0.019)
(0.018)
(0.016)
0.648**
0.492**
0.622**
0.660**
0.200**
0.159**
0.199**
0.266**
(0.050)
(0.056)
(0.064)
(0.051)
(0.024)
(0.030)
(0.028)
(0.026)
-0.094**
-0.035
-0.081**
-0.043**
-0.099**
-0.079**
-0.096**
-0.079**
(0.020)
(0.024)
(0.024)
(0.020)
(0.013)
(0.016)
(0.016)
(0.015)
Guildpaymem
-0.001
-0.004
-0.008**
-0.009**
-0.001
-0.001
-0.004**
-0.008**
ber (𝛽6 )
(0.001)
(0.002)
(0.002)
(0.002)
(0.001)
(0.001)
(0.001)
(0.002)
Guildlevel
0.059
0.007
0.115**
0.118**
0.038
0.058
0.056
0.110**
(𝛽7 )
(0.039)
(0.050)
(0.044)
(0.037)
(0.031)
(0.042)
(0.036)
(0.025)
Guildmoney
0.006
0.003
0.006**
0.022**
0.002
-0.003
0.005**
0.023**
(𝛽8 )
(0.005)
(0.003)
(0.002)
(0.008)
(0.002)
(0.002)
(0.001)
(0.004)
0.049
0.147
0.098
0.151*
0.047
0.018
0.061**
0.075**
(0.079)
(0.092)
(0.097)
(0.087)
(0.031)
(0.034)
(0.032)
(0.028)
-0.017**
-0.013**
-0.006
-0.008
-0.005**
-0.019**
-0.011**
-0.010**
(0.006)
(0.005)
(0.006)
(0.005)
(0.001)
(0.001)
(0.001)
(0.001)
0.034**
0.023**
0.007
0.018**
0.028**
0.021**
0.006
0.017**
(0.009)
(0.007)
(0.005)
(0.006)
(0.006)
(0.005)
(0.004)
(0.005)
-0.141**
-0.011
-0.006
-0.087**
-0.095**
-0.017
-0.013
-0.056**
(0.045)
(0.028)
(0.036)
(0.021)
(0.032)
(0.021)
(0.029)
(0.024)
27807
28371
34095
41620
29057
29937
35992
Level (𝛽2 )
Duration (𝛽3 )
Loginfrq (𝛽4 )
Questfrq (𝛽5 )
Isleader (𝛽9 )
Degree (𝛽10 )
Direct
influence
(𝜌1 )
Indirect
influence
(𝜌2 )
DIC
39791
**: p<0.05; *: p<0.1
22
Appendix A
Panel data model with individual fixed effect
MCMC estimation steps for Poisson regression model with network autoregression term
In the convenience of notation, we use 𝛏 to represent log(𝛌). Thus, the model is specified as follows:
y t ~ Poisson(exp(ξ t ))
ξ t  Xt β  θt +η  ε t
β ~ N (β 0 , D)
η~N (μ 0 , Γ 0 )
θt  1W1θt   2 W2θt  u t
ε t ~ N (0, I n )
u t ~ N (0,  2 I n )
 2 ~ IG ( s0 , q0 )
The posterior distribution of ξ conditional on the data is:
p(ξ t | β,σ 2 ,θt , Xt , y t )  p(ξ t | β,θt , Xt ) p(y t | ξ t )
(it   k 1 xitk  k  it  i )2 exp(it ) yit exp( exp(it ))
1
exp(
)
2
yit !
2
K
n

i 1
Step 1. Draw each 𝝃𝒊𝒕 with Metropolis-Hasting algorithm
itnew  itold  it
i ~ N (0,1)
The acceptance probability is given by
exp( exp(inew )  0.5  (itnew   k 1 xik  k  it  i ) 2  yititnew )
K
exp( exp(iold )  0.5  (itold   k 1 xik  k  it  i ) 2  yititold )
K
Step 2. Draw 𝜷, the prior distribution is 𝛽~𝑁(𝛽0 , 𝐷)
23
β |  2 , ξ t ,θt , η  N (μ β , Vβ )
T
μ β  Vβ ( XtT (ξ t  θt  η)  D1β 0 )
t 1
T
Vβ  (D1   XtT X t ) 1
t 1
D  400I n
β 0  (0,
, 0) '
Step 3. Draw 𝜽𝒕 , the prior distribution is 𝜽𝒕 ~𝑁(0, 𝟏)
θt | 1 ,  2 ,  2 , ξ t , η,β ~ N (μθ , Σθ )
B t ( 1 ,  2 )  (I  1W1,t   2 W2,t )
μθ  Σθ (ξ t  Xt β  η)
Σθ  (I 1   2 BtT Bt )1
Step 4. Draw 𝜼, the prior distribution is 𝜼~𝑁(𝝁𝟎 , 𝚪𝟎 ), 𝝁𝟎 = 𝟎, 𝚪𝟎 = 𝟏𝟎𝟎𝐈𝐧
η | ξ t , Xt , β, θt ,  2 ~ N (μ η , Γ η )
Γ η  (T I n   0 ) 1
T
μ η  Γ η ( (ξ t  Xt β  θt )  Γ0-1μ 0 )
t 1
Step 5. Draw 𝝈𝟐 , the prior distribution is 𝜎 2 ~𝐼𝐺(𝑠0 , 𝑞0 ), 𝑠0 = 5, 𝑞0 = 10
T

nT
 2 ~ IG (  s0 , t 1
2
θt T B t T B t θt
2
gamma distribution 𝑓(𝑥) =
 q0 )
𝛽 𝛼 𝛼−1 −𝛽𝑥
𝑥
𝑒
𝛤(𝛼)
Step 6. Draw 𝝆𝟏 , 𝝆𝟐 with Metropolis-Hasting algorithm
inew  iold  i，i ~ N (0,0.0052 )
The acceptance probability is given by
T
 | Bt ( inew ) |exp(
1
2 2
min( t T1
1
| Bt ( iold ) |exp( 2

2
t 1
T
 (θ
t 1
T
T
t
Bt ( inew )T Bt ( inew )θt ))
 (θtT Bt ( iold )T Bt ( iold )θt ))
t 1
Appendix B
24
,1)
Panel data model with individual random effect
MCMC estimation steps for Poisson regression model with network autoregression term
In the convenience of notation, we use 𝛏 to represent log(𝛌). Thus, the model is specified as follows:
y t ~ Poisson(exp(ξ t ))
ξ t  Xt β  θt +η  ε t
β ~ N (β 0 , D)
η~N (0 ,  2 )
0 ~ N ( 0 ,  2 )
0
  ~ IG (A, B)
2
θt  1W1θt   2 W2θt  u t
ε t ~ N (0, I n )
u t ~ N (0,  2 I n )
 2 ~ IG ( s0 , q0 )
The posterior distribution of ξ conditional on the data is:
p(ξ t | β,σ 2 ,θt , Xt , y t )  p(ξ t | β,θt , Xt ) p(y t | ξ t )
(it   k 1 xitk  k  it  i )2 exp(it ) yit exp( exp(it ))
1
exp(
)
2
yit !
2
K
n

i 1
Step 1. Draw each 𝝃𝒊𝒕 with Metropolis-Hasting algorithm
itnew  itold  it
i ~ N (0,1)
The acceptance probability is given by
exp( exp(inew )  0.5  (itnew   k 1 xik  k  it  i ) 2  yititnew )
K
exp( exp(iold )  0.5  (itold   k 1 xik  k  it  i ) 2  yititold )
K
Step 2. Draw 𝜷, the prior distribution is 𝛽~𝑁(𝛽0 , 𝐷)
25
β |  2 , ξ t ,θt , η  N (μ β , Vβ )
T
μ β  Vβ ( XtT (ξ t  θt  η)  D1β 0 )
t 1
T
Vβ  (D1   XtT X t ) 1
t 1
D  400I n
β 0  (0,
, 0) '
Step 3. Draw 𝜽𝒕 , the prior distribution is 𝜽𝒕 ~𝑁(0, 𝟏)
θt | 1 ,  2 ,  2 , ξ t , η,β ~ N (μθ , Σθ )
B t ( 1 ,  2 )  (I  1W1,t   2 W2,t )
μθ  Σθ (ξ t  Xt β  η)
Σθ  (I 1   2 BtT Bt )1
Step 4. Draw 𝜼, the prior distribution is 𝜼~𝑁(𝜂0 , 𝜎𝜂2 )
η | ξ t , Xt , β, θt ,  2 ~ N (μ η , Γ η )
Γ η  (T I n   2 I n ) 1
T
μ η  Γ η ( (ξ t  Xt β  θt )   20 1n )
t 1
Step 5. Draw 𝜂0 , the prior distribution is 𝜂0 ~N(𝜇0 , 𝜎𝜇20 )
0 | η,  2 ~ N (  ,  2 )
0
0
   (   n  )
2
2
0
2 1
0
    ( 2 0   2 1n' η)
2
0
0
0
Step 6. Draw 𝜎𝜂2 , the prior distribution is 𝜎𝜂2 ~𝐼𝐺(𝐴, 𝐵)
n
2
2 ~ IG(  A,
( η  0 1n ) '( η  0 1n )
2
 B)
Draw 𝜎 2 , the prior distribution is 𝜎 2 ~𝐼𝐺(𝑠0 , 𝑞0 ), 𝑠0 = 5, 𝑞0 = 10
T
 2 ~ IG (
nT
 s0 ,
2
θ
t 1
T
t
B t T B t θt
2
26
 q0 )
gamma distribution 𝑓(𝑥) =
𝛽 𝛼 𝛼−1 −𝛽𝑥
𝑥
𝑒
𝛤(𝛼)
Step 6. Draw 𝝆𝟏 , 𝝆𝟐 with Metropolis-Hasting algorithm
inew  iold  i，i ~ N (0,0.0052 )
The acceptance probability is given by
T
| B (
1
θtT Bt ( inew )T Bt ( inew )θt )
2 2
min(
,1)
1
old
T
old T
old
| Bt ( i ) | exp( 2 θt Bt ( i ) Bt ( i )θt )

2
t 1
t 1
T
t
new
i
) | exp(
27